An entire infinite plane fit inside a circle — Escher's favorite geometry
The Poincare disk model maps the entire infinite hyperbolic plane into the interior of a unit circle. Distances grow without bound as you approach the boundary, and straight lines (geodesics) appear as circular arcs that meet the boundary at right angles. This conformal model preserves angles, making it ideal for understanding hyperbolic geometry visually — and it was the model that inspired M.C. Escher's famous Circle Limit woodcuts.
In the Poincare disk, geodesics are circular arcs perpendicular to the boundary circle. Click pairs of points inside the disk to draw geodesics. Notice how arcs that look “bent” in Euclidean terms are actually the shortest paths in hyperbolic space.
Try it: Draw several geodesics through the same point and observe how they fan out. Try drawing geodesics near the boundary — they become nearly tangent to the circle, reflecting the infinite distance to the edge.
Drag two points inside the disk to compare Euclidean distance |p - q| with hyperbolic distance d(p, q). Near the center, the two metrics nearly agree. Near the boundary, a tiny Euclidean step corresponds to an enormous hyperbolic distance — the metric blows up as 2/(1 - r²).
Drag point A or B inside the disk. Concentric contours show lines of constant hyperbolic distance from A. The dashed line shows the Euclidean straight path; the solid arc is the hyperbolic geodesic.
Classical Euclidean constructions have hyperbolic counterparts. Use this tool to perform geodesic constructions: draw geodesics, construct hyperbolic circles, find geodesic midpoints, and draw perpendicular bisectors — all computed exactly in the Poincare disk model.
Key insight: The perpendicular bisector of a hyperbolic segment is the set of all points equidistant from both endpoints — just as in Euclidean geometry. But the bisector itself is a geodesic arc, not a straight line.
M.C. Escher's Circle Limit series rendered a hyperbolic tiling with repeating motifs that shrink toward the boundary — each motif the same hyperbolic size, but appearing smaller in the Euclidean picture. This GPU-accelerated shader lets you explore {p, q} tilings with different motifs, color palettes, and automatic Mobius rotation.
Escher Circle Limit — A GPU-rendered {p,q} hyperbolic tiling. The motifs are congruent in hyperbolic geometry but shrink in the Euclidean picture as they approach the boundary circle. Change p and q to explore different tilings (valid when 1/p + 1/q < 1/2).